Baden on "[Plugin: T(-) Countdown] Timer displays briefly on page load, then disappears"

Just tested this and it does work on a clean install.
This leads me to believe you have another plugin or theme element that is hijacking the jQuery datetime picker.

But this is clearly the issue. On the front end the target date is:
'day': 01, 'month': 01, 'year': 1970, 'hour': 00, 'min': 00, 'sec': 00,
Can you disable all other plugins and see if it works then?
if so, re-enable your plugins one by one until the date slider breaks, and let us know the offending plugin.

For a work around, you should be able to manually type in the date...
Fret not, we'll help get you sorted out in short order.